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hep-th/0412243HUTP-04/A034

Khovanov-Rozansky Homology and Topological Strings

Sergei Gukov1, Albert Schwarz2, and Cumrun Vafa1

1 Jefferson Physical Laboratory,

Harvard University,

Cambridge, MA 02138, USA

2 Department of Mathematics,

University of California,

Davis, CA 95616, USA

Abstract

We conjecture a relation between the sl(N) knot homology, recently introduced by

Khovanov and Rozansky, and the spectrum of BPS states captured by open topological

strings. This conjecture leads to new regularities among the sl(N) knot homology groups

and suggests that they can be interpreted directly in topological string theory. We use this

approach in various examples to predict the sl(N) knot homology groups for all values of

N . We verify that our predictions pass some non-trivial checks.

Dedicated to the memory of F.A. Berezin.

December 2004

1. Introduction and Summary

During the past twenty years, topological field theories have been the source of the vig-

orous interaction between theoretical physics and pure mathematics, increasingly fruitful

for both fields. One of the famous examples of topological field theories is a Chern-Simons

gauge theory [1,2]. Observables in this theory are naturally associated with knots and

links. Specifically, given an oriented knot, K, and a representation of the gauge group, R,

one can construct a Wilson loop operator, WR(K), whose expectation value turns out to

be a polynomial invariant of K, such as the Jones polynomial and its generalizations [2].

Here, we will be mainly interested in the case where R = is the fundamental

representation of sl(N). The corresponding quantum invariant

PN (q) = 〈W (K)〉 (1.1)

is a one-variable specialization of the HOMFLY polynomial [3]. It can be determined by

the sl(N) skein relation

qNPN (L+)− q−NPN (L−) = (q−1 − q)PN (L0) (1.2)

and by the normalization

PN (unknot, q) = [N ] =qN − q−N

q − q−1(1.3)

L L L+ 0 −

Fig. 1: Link diagrams connected by the skein relation.

As shown in [4], Chern-Simons theory can be embedded in string theory. Later it was

shown [5,6,7] using highly non-trivial ‘stringy dualities’ that this leads to a reformulation

of quantum sl(N) knot invariants in terms of topological string amplitudes (Gromov-

Witten invariants). Moreover, it was realized that the polynomial invariants PN (q) can be

1

reformulated in terms of integers N ,Q,s which capture the spectrum of BPS states in the

string Hilbert space1 [7,8]:

PN (q) =1

q − q−1

∑

s,Q∈ZZ

N ,Q,sqNQ+s (1.4)

It is difficult to give a mathematically rigorous definition of BPS degeneracies. In the

simpler case of closed topological strings, there is a similar notion of BPS degeneracies

[6]. Attempts to give it a precise mathematical definition [9,10,11,12,13] led to a deeper

understanding of Gromov-Witten invariants, though even there not all questions are an-

swered. In the open string case, even though a mathematically rigorous proof of (1.4) is not

available this equation is strongly supported by physical arguments and highly non-trivial

checks, in particular, by the fact that considering it as a definition of N ,Q,s one always

obtains integers.

In another line of development, the quantum sl(N) invariant PN (q) was lifted to a

homological knot invariant [14,15,16,17]. For a given knot K and a fixed value of N , the

sl(N) homological invariant is a doubly graded cohomology theory, Hi,jN (K), whose Euler

characteristic with respect to one of the gradings equals the quantum sl(N) invariant,

PN (q) =∑

i,j∈ZZ

(−1)iqj dimHi,jN (K) (1.5)

Following [18], we also introduce the graded Poincare polynomial,

KhN (q, t) :=∑

i,j∈ZZ

tiqj dimHi,jN (K) (1.6)

Relegating technical details to the following sections, let us summarize some important

features of HN :

i) HN is a functor (from the category of links and cobordisms to the category of vector

spaces [16,18,19,20]);

ii) HN is stronger than PN , at least in the case N = 2;

iii) HN is hard to compute (at present, only the sl(2) homological invariant has been

computed for knots with small number of crossings [18,21]);

iv) HN cries out for a physical interpretation!

1 In our normalization, the integer invariants N ,Q,s are non-trivial only for even values of s.

2

In this paper, we take a modest step towards understanding the physical interpre-

tation of the sl(N) knot homology by relating it to the BPS spectrum of states in open

topological strings. It naturally leads us to new regularities among the homology groups,

Hi,jN (K), which we hope may ultimately lead to a better understanding of the sl(N) ho-

mological invariant on the deep conceptual level. In particular, we will be interested in

the dependence of the homology groups Hi,jN (K) on the rank of sl(N). By analogy with

(1.4), we expect that, at least for sufficiently large values of N , the following complex has

a very simple structure:

H•N{+1} − H•

N{−1} (1.7)

where {n} denotes the shift in q-grading up by n. Specifically, we propose the following:

Conjecture: For a knot K and sufficiently large2 values of N , the graded Poincare poly-

nomial of the doubly graded complex (1.7) can be expressed in terms of N -independent

integer invariants DQ,s,r(K),

(q − q−1)KhN (q, t) =∑

Q,s,r∈ZZ

DQ,s,rqNQ+str

(1.8)

Notice, this conjecture implies that (for sufficiently large N) the set of values of the grading

i in Hi,jN (K) is contained in a set that does not depend on N . There is also a generalization

of this conjecture for links, see below.

The integer invariants DQ,s,r provide a refinement of the open string BPS degeneracies

by decomposition of the states in terms of an additional U(1) charge (grading) r in a sense

that

N ,Q,s =∑

r∈ZZ

(−1)rDQ,s,r. (1.9)

This is parallel to the refinement of BPS degeneracies for closed strings found in [22].

Notice, that the conjectured relation (1.8) is manifestly consistent with eqs. (1.4) - (1.6)

and the relation (1.9).

Organization of the Paper

In section 2, we briefly review some of the main statements about integer BPS degen-

eracies for open strings and the sl(N) knot homology. The more detailed physical setup

2 It turns out, however, that for many simple knots this conjecture works for all N ≥ 2.

3

and the interpretation of the sl(N) knot homology as BPS degeneracies DQ,s,r is discussed

in section 3. In section 4, we study explicit examples. In particular, we use (1.8) to pre-

dict the sl(N) homological invariants for certain knots with small number of crossings and

verify that our predictions pass some non-trivial checks. In section 5, we discuss various

generalizations of the above conjecture. Finally, in the appendices we present the explicit

form of the HOMFLY polynomial and the sl(2) homological invariant for some knots.

2. Preliminaries

2.1. Conventions

Here we follow the standard conventions in knot theory, where the unreduced Jones

polynomial has expansion in integer powers of q, J ∈ ZZ[q, q−1]. For example, in these

conventions, the quantum sl(N) invariant of the trivial knot has the form (1.3). These

conventions are compatible with the ones used in [14,15,16,17,18], but differ from the

conventions used in the physics literature [5,6,7,8,23,24]. The two sets of conventions are

related by a simple change q → q1/2. We also warn the reader that in some literature our

variable q is denoted q−1 and, similarly, t is denoted t−1.

Throughout, we work overQ. In particular, all cohomology groups areQ-vector spaces,

and dim(Hi,jN ) denotes dimQ(Hi,j

N ).

2.2. A Brief Review of the BPS Degenaracies and Knot Invariants

The generating function of Wilson loop (knot) observables in Chern-Simons theory

can be written in terms of the f -polynomials [7]:

FCS(V ) =∞∑

d=1

∑

R

1

dfR(q

d, λn)TrRVd (2.1)

where λ := qN and N is the rank of the gauge group. The polynomials fR(q, λ) can be

expanded in q and λ [7,8]

fR(q, λ) =1

q − q−1

∑

s,Q

NR,Q,sλQqs =

=∑

g≥0

∑

Q

∑

R′,R′′

CRR′R′′SR′(q)NR′′,g,Q(q−1 − q)2g−1λQ

(2.2)

4

In this expression, R, R′, and R′′ denote representations of the symmetric group, which

we can label by Young tableaus with ℓ boxes. The coefficients CRR′R′′ are the Clebsch-

Gordan coefficients of the symmetric group, and the function SR(q) is non-zero only for

hook representations of the form

(2.3)

Specifically, if R is a hook representation with ℓ − d boxes in the first row, then SR(q) =

(−1)dq−ℓ−1

2+d, and SR(q) = 0 otherwise. Finally, the numbers NR,Q,s and NR,g,Q encode

the BPS spectrum for open topological strings.

The integer BPS degeneracies can be defined in terms of cohomology computations on

certain spaces. In particular, for the fundamental represenation, N ,g,Q can be defined as

the Euler characteristics of the cohomology of moduli spaces3 Mg,Q. We refer the reader

to the original papers [6,7,8] for the details, and briefly sketch here the results that will be

relevant to us in what follows. There is an explicit expression for N ,g,Q in terms of the

cohomology of Mg,Q,

N ,g,Q = ǫ χ(Mg,Q) (2.4)

where4 ǫ = ±1.

In the case we are considering, R = , the polynomial f (q, λ) is the unnormalized

HOMFLY polynomial. Its value at λ = qN gives the sl(N) polynomial PN (q):

PN (q) = f (q, qN) (2.5)

The polynomial f (q, λ) has a simple representation in terms of NR,g,Q:

f (q, λ) =∑

Q

∑

g≥0

N ,g,Q(q−1 − q)2g−1λQ (2.6)

Comparing this formula with eq. (2.2), we obtain an expression for the integers N ,Q,s in

terms of N ,g,Q,

N ,Q,s = −∑

g≥0

(−1)g+s/2

(2g

g + s/2

)N ,g,Q (2.7)

Using (2.4), we can further express N ,Q,s via dimensions of the cohomology groups

Hk(Mg,Q),

N ,Q,s = −ǫ∑

g≥0,k

(−1)k+g+s/2

(2g

g + s/2

)dimHk(Mg,Q) (2.8)

3 As discussed in section 3 this is the moduli space of holomorphic curves of genus g with one

boundary ending on a suitable Lagrangian submanifold.4 As discussed in [7] the origin of the sign ǫ is related to the analytic continuation one has to

make in order to relate topological string amplitudes with the quantum-group knot invariants.

5

Example. The Unknot

For the unknot the corresponding moduli spaces are isolated points, so that

Hk(Mg,Q) ={Q if k = g = 0 and Q = ±1

0 otherwise(2.9)

Hence, from (2.4) we find that the only non-zero invariants are [7]:

N ,0,Q=±1 = ∓1 (2.10)

This leads to the usual expression for the HOMFLY polynomial

f (unknot) =λ− λ−1

q − q−1=

qN − q−N

q − q−1(2.11)

Let us consider another example.

Example. The Trefoil Knot

In this case, the non-zero BPS invariants are [8]:

N ,0,1 = 2, N ,0,3 = −3, N ,0,5 = 1,

N ,1,1 = 1, N ,1,3 = −1(2.12)

Substituting this into (2.6), we find

f (31) =2λ− 3λ3 + λ5

q−1 − q+ (λ− λ3)(q−1 − q) (2.13)

Notice, that in the rank 2 case, λ = q2, we recover the usual expression for the unreduced

Jones polynomial of the trefoil knot,

J(31) = q + q3 + q5 − q9 (2.14)

2.3. A Brief Review of the sl(N) Knot Homology

In a fascinating work [14], Khovanov introduced a new homological knot invariant,

which has the Jones polynomial as its graded Euler characteristic. Subsequently, this work

was extended to a categorification of the quantum sl(3) invariant [16] and, more recently,

to a categorification of the quantum sl(N) invariant [17]. Although these constructions

differ in details, the basic idea is to associate a chain complex of graded vector spaces,

6

C•N (L), to a plane diagram of a link L colored by a fundamental representation of sl(N).

The bigraded cohomology groups of this complex,

HN (L) := H∗(C•N ) (2.15)

do not depend, up to isomorphism, on the choice of the projection of L. It is also convenient

to define the graded Poincare polynomial,

KhN (L) :=∑

i,j∈ZZ

tiqj dimHi,jN (L) (2.16)

and the graded Euler characteristic,

χq(L) :=∑

i,j∈ZZ

(−1)iqj dimHi,jN (L) (2.17)

One of the main results in [14,16,17] states that χq(L) is equal to the quantum sl(N)

invariant of L:

PN (q) = χq(L) = KhN (L)|t=−1 (2.18)

Notice, that since the Euler characteristic of the cohomology Hi,jN (L) is the same as the

Euler characteristic of the chain complex Ci,jN (L) itself, we can write (2.17) - (2.18) as

PN (q) =∑

i,j∈ZZ

(−1)iqj dim Ci,jN (L) (2.19)

Example. The Trefoil Knot

For the trefoil knot and N = 2, the Khovanov’s invariant KhN=2(L) has the

following form:

KhN=2(31) = q + q3 + q5t2 + q9t3 (2.20)

It is easy to see that at t = −1 we recover the usual Jones polynomial (2.14) of the

trefoil knot.

Now, let us say a few words about the structure of the complex C•N (L). Let W =

⊕mWm be a graded vector space. Its graded dimension is

dimq W =∑

m

qm dimWm (2.21)

7

In the spirit of the derived categories, we will be considering chain complexes of these

graded vector spaces, e.g.

C• : . . . −→dr−1

Cr −→dr

Cr+1 −→dr+1

Cr+2 −→dr+2

. . . (2.22)

where r is the “height” of the graded vector space Cr.

Let us also introduce two translation functors: the degree shift

W{l}m := Wm−l (2.23)

and the height shift

C[s]r := Cr−s (2.24)

In other words, C•[s] denotes the complex C• shifted s places to the right. Also, note that

(2.23) implies

dimq W{l} = ql dimq W (2.25)

=

1 1

1 1

2

Fig. 2: A planar trivalent graph Γ near a “wide edge” and its representation

in the Murakami-Ohtsuki-Yamada terminology [25], where each oriented edge is

labelled by a fundamental weight of sl(N).

For each value of N , there is a combinatorial algorithm to construct the ZZ⊕ZZ-graded

chain complex C•N (L) from a plane diagram D of the link L. Namely, using a suitable skein

relation one takes all possible resolutions of the link diagram D,

C•N (L) =

⊕

Γ

C•N (Γ) (2.26)

where each resolution Γ is a planar trivalent graph, as in fig. 2. For example, in the sl(2)

case, each Γ is a collection of plane cycles, and a simple algorithm for constructing C•2 (L)

was proposed by Khovanov in the original paper [14]; essentially, one has

C•2 (Γ)

∼= V ⊗#(cycles) (2.27)

8

Here, V is a cohomology ring of a 2-sphere, V ∼= H∗(S2), so that dimq(V ) = q + q−1.

Similarly, in the sl(3) case, one can systematically construct C•3 (L) using web cobordisms

(“foams”) [16].

A general algorithm, which in principle5 allows to construct the sl(N) knot homology

for arbitrary N , was proposed recently by Khovanov and Rozansky [17]. In this approach,

C•N (L) is constructed as a ZZ ⊕ ZZ ⊕ ZZ2-graded complex of Q-vector spaces. It turns out

that the cohomology groups of this complex are non-trivial only for one value of the ZZ2

grading, so that H∗(C•N ) is ZZ ⊕ ZZ-graded as in (2.15). Basically, a 2-periodic complex

C•N (L) is a tensor product of matrix factorizations:

M0 −→d0

M1 −→d1

M0 (2.28)

We remind that a matrix factorization of a homogeneous potentialW (xi) is a collection

of two free modules M0, M1 over the ring of polynomials in xi, and two maps d0 : M0 →

M1 and d1 : M1 → M0, such that

d0d1 = W · Id , d1d0 = W · Id (2.29)

Choosing a basis in M0, M1 we can think of d0 and d1 as matrices with polynomial entries.

One can also combine d0 and d1 into an odd matrix of the form

Q =

(0 d1

d0 0

)(2.30)

so that (2.29) can be written in a compact form

Q2 = W · Id (2.31)

In other words, Q can be regarded as an odd endomorphism (also known as “twisted differ-

ential”) acting on a free ZZ2-graded moduleM = M0⊕M1 over R =Q[xi]. When, following

Khovanov and Rozansky [17], one takes a tensor product over matrix factorizations of the

form (2.28), the W ’s cancel, so that Q becomes an ordinary differential, Q2 = 0, and the

result becomes a complex,

C•N (L) = ⊗i Mi (2.32)

5 In practice, the previous two constructions remain more suitable for explicit calculations of

the sl(2) and sl(3) knot homology.

9

A factorization M is called trivial (or contractible) if it has the form

R −→1 R −→W R (2.33)

Dividing the category of matrix factorizations (2.28) by the equivalence relation

M ∼ M ⊕Mtrivial (2.34)

we get a “derived” category of factorizations of W , denoted by hmfW in [17], whose

isomorphisms are morphisms which induce isomorphisms in cohomology. Furthermore,

this category has a triangulated structure, which plays an important role in the notion

of D-brane stability, see [26] for a recent review. In fact, the category hmfW is a simple

example of a “D-brane category”, whose objects are D-branes in N = 2 Landau-Ginzburg

model with a superpotential W [27,28,29,30]. In particular, with each “twisted complex,”

Mi ∈ hmfWi, we can associate a Landau-Ginzburg model and a D-brane in it. The

tensor product of these Landau-Ginzburg models is a two-dimensional N = 2 theory, cf.

[31,32,33,34]:

⊗i LGWi(2.35)

We refer the reader to [14,16,17,18] for further details on the construction of the sl(N)

knot homology.

3. Physical Interpretation of the sl(N) Knot Cohomology

In this section we start by briefly reviewing6 the relations between BPS states and

topological strings. We then discuss how Chern-Simons theory on M is embedded in open

topological strings on T ∗M . We also review how in the case of M = S3 ‘large N’ stringy

dualities relate it to closed topological string on a deformed geometry (the small resolution

of the nodal singularity). Also reviewed is how the HOMFLY polynomial invariants can

be formulated in this context. We then go on to define a refined version of the BPS

degeneracies which leads to a refinement of open topological string amplitudes. We then

propose a relation between this refined BPS degeneracies to the sl(N) knot homology

which is a categorification of quantum sl(N) polynomial invariants.

6 For a more detailed review on these topics the reader can consult [35].

10

3.1. Topological Strings and BPS Degeneracies

Closed topological string, which encodes Gromov-Witten invariants, deals with holo-

morphic maps from closed Riemann surfaces of arbitrary genus to Calabi-Yau threefolds

X . The partition function Z is a function of the Kahler moduli of Calabi-Yau threefold,

which we denote by T and of the string coupling constant gs. The dependence on T arises

by weighing each map by exp(−A) where A is the area of the curve. The gs dependence

enters by weighing a genus g Riemann surface by the factor g2g−2s . It has been argued in

[6] that Z can be rewritten in terms of integers nkQ as

Z(T, gs) =∏

k,Q

[ ∞∏

n=1

(1− q2n+2ke−〈Q,T 〉)n]nk

Q

where q = e−gs/2 and Q denotes an element of H2(X,ZZ). The integers nkQ denote the

degeneracy of BPS states. More precisely these correspond to embedded holomorphic

curves in the class Q and the index k is related to the genus of the curve. Physically

these degeneracies are computed by studying the moduli spaces of D-branes in the class

Q. Note that the moduli space includes the choice of a flat connection on the D-brane

in addition to the moduli associated to moving the D-brane inside X . Let us consider

an idealized situation (the more general case was discussed in [11]; see also [9,10,12,13]):

Suppose that as we vary the curve inside X its genus does not change (and assume there

are no monodromies of the cycles of the Riemann surface). In this case the moduli space

naturally splits into the moduli space of the flat connection, which is a 2g dimensional

torus T 2g, and that of the geometric moduli MQGeom.:

MQ = T 2g ×MGeom.

In this simple situation one finds that

∑

k

nkQq

k = χ(MGeom.) · (q − q−1)2g

Here the factor (q−q−1)2g corresponds to the Poincare polynomial of T 2g. The reader may

wonder why the Poincare polynomial for the full space MQ does not enter the topological

string amplitudes, and only the Euler characteristic of the geometric moduli enters? The

reason is that as one changes the complex moduli of Calabi-Yau X , it turns out that

MGeom. changes topology (and even dimension!), but its Euler characteristic does not

change. So only χ is an invariant of the Calabi-Yau X .

11

However, there are cases where X is rigid and has no complex moduli. In such

cases one could have obtained a more refined invariant by introducing a new parameter

t which captures the Poincare polynomial PMGeom(t) of the geometric moduli. In the

physical context this corresponds to an addition ‘charge’ that we can measure and refine

the degeneracies to nk,k′

Q . This extra charge appears as follows: In compactifications on

M-theory on Calabi-Yau 3-folds we end up with a four dimensional space IR4. The rotation

symmetry group is thus SO(4) = SU(2)L ×SU(2)R. The charge measured by powers of q

(together with (−1)F ) corresponds to U(1)L ∈ SU(2)L. We could also measure the charge

U(1)R ∈ SU(2)R. So we would have considered

∑

k,k′

nk,k′

Q q2ktk′

= (q − q−1)2gPMGeom(t)

In fact in view of the application to the knots it is natural to redefine the basis of the

U(1)L × U(1)R charges. This is because in that case we have to introduce Lagrangain

D-branes which fill IR2 ⊂ IR4. We still have a U(1)⊗2 symmetry. However one of the

U(1)’s corresponds to the physical spin in the subspace IR2 and is given by the diagonal

in U(1)L × U(1)R. We will thus redefine7

q → iqt12

and write

PQ(q, t) =∑

k,k′

nk,k′

Q (q2t)k(t/q2)k′

= (−1)g(qt12 + q−1t−

12 )2gPMGeom

(t) (3.1)

This structure of course presupposes that the moduli space MQ has a product structure.

This is seldom the case. One case where this happens naturally is when the geometric

moduli space is a point. One may think that in the more general case the splitting of

the Poincare polynomial to two variable polynomials may not exist. However physical

reasoning, i.e. the existence of two U(1) charges that one can measure, guarantees that

this must be possible even if the moduli space does not split. Moreover if X is rigid it is an

invariant for the Calabi-Yau. Examples of this have already been computed and checked

against physical computations in [22], see also [10,36] for a mathematical discussion. This

7 The choice of i =√−1 and also the factor of 1/2 in the exponent of t in the formula is for

convenience of comparison with the conventions used in the knot theory literature.

12

means that PQ(q, t) always exists but it does not in general take the simple form given

above as a product of two prefactors.

One can also consider open topological strings. To do this we consider Lagrangian

subspaces Li of X with some multiplicity Ni associated to each. We then consider holo-

morphic maps from Riemann surfaces with arbitrary boundaries such that the boundaries

all lie on some Lagrangian brane Li. Each such configurations will receive a factor Ni;

or if the boundary circle wraps over the cycle S1 r times, with holonomy factor Ui, we

pick up a factor of trUri . The open topological string amptitudes can again be recast in

terms of the BPS degeneracies, D2 branes embedded in X which can have a boundary

on the Lagrangian submanifold. The degeneracies will also have a label of Representation

of the brane, encoding different ways the D2 branes end on the Lagrangian submanifold.

The story in this case is somewhat more complicated that in the closed string case [7,8] .

However just as in the closed string case one expects the counting of these degeneracies to

depend on choice of charge Q in the relative homology H2(X,Li), the spin s and in addi-

tion on the representation label R, NR,Q,s, which is related to cohomology computations

on the moduli space of D2 branes with boundaries (again including the flat connection).

The open topological string amplitudes can be written in terms of NR,Q,s as discussed in

(2.1).

What we would like to ask now is whether we can define a more refined invariant even

in this case, in other words, are there more physical charges we can measure? Let us recall

how the open string BPS charges get embedded in the closed string charges [7,8]: As noted

above we have Lagrangian D-branes filling IR2 ⊂ IR4 and thus in the open string context

we still have both U(1) symmetries. Let us denote these two charges by (s, r), where s

denotes the spin in IR2. Thus we expect that any degeneracies can also be further refined

according to these two charges, in particular we would get new degeneracies DR,Q,s,r which

have the property that

NR,Q,s =∑

r

(−1)rDR,Q,s,r (3.2)

One could ask if the integers DR,Q,s,r are invariant, i.e. do they depend on moduli of

Calabi-Yau or moduli of the brane? It is easy to see, just as in the closed string case, that

they could depend on the complex structure moduli of the Calabi-Yau. But they cannot

depend on the Kahler moduli of the Calabi-Yau, or the moduli of the brane. The latter

statement follows from the fact that the BPS states are in the (c, c) multiplets whereas

the Kahler moduli of Calabi-Yau or the brane moduli are in the (a, c) multiplets and so it

13

cannot give mass to the (c, c) multiplets. In particular if we are dealing with a rigid Calabi-

Yau we would find the more refined invariants DR,Q,s,r, similar to what we discussed for

the closed string BPS states.

In this paper for simplicity we will mainly concentrate on the case where R = , i.e.,

N ,Q,s and the corresponding refinement will be denoted by DQ,s,r. For the representation

R = the only relevant BPS brane are Riemann surfaces with one boundary on the

Lagrangian submanifold. Suppose we have an isolated Riemann surface of genus g with

one boundary. Then, just as we discussed in the closed string case we would find the

refined Poincare polynomial:∑

s,r

DQ,s,rqstr = (−1)g(qt

12 + q−1t−

12 )2g (3.3)

3.2. Chern-Simons Theory and Open Topological Strings

It was shown in [4] that U(N) Chern-Simons theory on a three manifold M can be

reformulated in terms of open topological string on the Calabi-Yau manifold X = T ∗M ,

where we consider N D-branes wrapping the Lagrangian cycle M . What this means is

that we consider Riemann surfaces with boundaries and ‘count’ holomorphic maps from

the Riemann surface onto T ∗M with the condition that the boundary ends on M , and

that each boundary gets a factor of N corresponding to which brane it ends on. Moreover

the Chern-Simons coupling 2πi/(k + N) gets identified with topological string coupling

constant gs. In fact there are no honest holomorphic maps from Riemann surfaces to T ∗M

however we end up with degenerate maps which approach a holomorphic map. These

degenerate maps correspond to ribbon graphs on M and in fact reproduce the Feynman

graphs of the Chern-Simons perturbation theory.

3.3. Open Topological String on S3 and a Large N Duality

It was shown in [5] that for the case of M = S3, with N branes wrapping S3 there

is an equivalent description of the open topological strings in terms of a closed topological

string. The corresponding Calabi-Yau is obtained by a geometric transition: S3 shrinks

to a point, leading to conifold which can be smoothed out by a small blow up leading

to a Calabi-Yau geometry which is a total space of the bundle O(−1) ⊕ O(−1) over P1.

Moreover the Kahler moduli of P1 (i.e. its area) is T = Ngs = 2πiN/k + N . In other

words

exp(−T/2) = λ = qN

This therefore gives a relation between Chern-Simons theory on S3 and closed topological

strings on the resolved conifold.

14

3.4. Introducing the Knots

One can extend the above relation between Chern-Simons theory on S3 and topological

strings on resolved conifold, to include knot observables [7]: Consider a knot K in S3. K

defines a non-compact Lagrangian submanifold LK in T ∗S3 given by the conormal bundle.

If we wrap M D-branes on LK we also get a Chern-Simons theory on it. We have two

Wilson loop observables U, V on S3 and LK . Viewing V as fixed, wrapping branes on LK

deforms the Chern-Simons theory on S3 and gives a new partition function:

Z(V,N, k) =∑

R

trRV 〈trRU〉 =∑

R

(trRV )WR(K)

The large N duality relates this to computation of topological strings on the resolved

conifold where the branes wrapping S3 have disappeared, but the branes wrapping LK

continue to be there. The fact that there should be such a Lagrangian LK even after the

transition has been shown in [37]. This means that we can rewrite Z(V,N, k) in terms of

open topological strings in this new geometry. The N and k dependence capture the Kahler

moduli dependence and the string coupling constant, and the V dependence captures how

the Riemann surface ends on LK (if a boundary wraps around the S1 ⊂ LK r times it

leads to a factor trV r). In this case the Calabi-Yau is rigid and so we can define the refined

invariants DR,Q,s,r associated to each knot K. For the case of R = it is natural to expect

this is related to sl(N) knot homology. Let us try to see if we can make this connection

more precise.

From the definition of open topological string amplitude and its relation to quantum

sl(N) knot invariant (1.4) it follows that

PN (q) =1

q − q−1

∑

Q,s,r∈ZZ

(−1)rqNQ+sDQ,s,r (3.4)

where we also used (3.2). It is tempting in the above formula to replace (−1) by a new

variable t and write the generating function of the refined BPS degeneracies DQ,s,r in the

form1

q − q−1

∑

Q,s,r∈ZZ

DQ,s,rqNQ+str (3.5)

We expect, that there is a similar expression for the graded Poincare polynomial of the

sl(N) homological invariant in terms of some integers DQ,s,r, cf. (1.8):

KhN (q, t) =1

q − q−1

∑

Q,s,r∈ZZ

DQ,s,rqNQ+str (3.6)

15

where DQ,s,r obey

N ,Q,s =∑

r∈ZZ

(−1)rDQ,s,r (3.7)

The above statement, if true, would give a highly non-trivial prediction for the N -

dependence of the homological sl(N) invariants. In fact, we find this dependence is satisfied

in all the examples we have checked, as will be discussed in the next section.

Although we do not claim that the generating functions (3.5) and (3.6) need to be

equal, we expect a simple relation between the refined BPS invariants, DQ,s,r, and their

physical counterparts, DQ,s,r. Indeed, in the examples considered below, one can recognize

contributions of genus-g curves of the form

∑

s,r

DQ,s,rqstr = (−1)g(qt

12 + q−1t−

12 )2gtαKQ (3.8)

where αK is some simple invariant of the knot K. Comparing this expression with (3.1)

and (3.3), one can interpret such terms either as contributions of isolated curves, in which

case PMGeom(t) = 1, or as contributions8 of genus-g curves with PMGeom

(t) = tαKQ. In

the first case, we expect the extra factor of tαKQ to come from the change of basis for

(Q, s, r) in the relation between DQ,s,r and DQ,s,r:

DQ,s,r = DQ,s,r−αKQ (3.9)

This change of basis would be very natural in a relation between two graded homology

theories. Notice, it does not affect the N -dependence of (3.5) - (3.6) which is true even

without such a shift. It would be important to understand this change of basis more deeply.

It is tempting to speculate that this is related to replacing the sl(N) knot homology with

a gl(N) version, as the corresponding Chern-Simons theory related to topological strings

is the U(N) version and not the SU(N) version.

On the other hand, if we interpret (3.8) as a contribution of genus-g curve with

PMGeom(t) = tαKQ, there is no need for the change of basis r → r − αKQ, and we can

simply identify DQ,s,r = DQ,s,r.

8 Such contributions can arise if there is a non-trivial monodromy.

16

4. Examples

4.1. The Unknot

The conjecture (1.8) automatically holds for the unknot, the only knot for which all

the sl(N) knot homology groups are known at present.

Indeed, for the unknot, the sl(N) knot cohomology coincides with its graded Euler

characteristic (1.3) (since all the non-trivial groups Hi,jN (unknot) are only in degree i = 0).

Therefore, the graded Poincare polynomial manifestly obeys the relation (1.8),

KhN (q, t) =1

q − q−1

∑

Q,s,r

DQ,s,rqNQ+str

with non-zero invariants

D−1,0,0 = −1, D1,0,0 = 1 (4.1)

Notice, that, in the case of the unknot, the moduli spaces Mg,Q are isolated points when

Q = 1 and Q = −1, cf. (2.9). These are precisely the values of Q for which we find

non-trivial invariants (4.1). Moreover, for each value of Q, the invariants (4.1) have the

structure (3.3) consistent with a contribution of an isolated curve with g = 0.

Let us consider the explicit form of the sl(N) cohomology groups for the unknot. For

example, for N = 4 we have

Hi,jN=4 :

0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 Q 0 Q 0 Q 0 Q 0 0 0

(4.2)

where the vertical direction represents the index i, while the horizontal direction represents

the grading j. In general, the complex H•N (unknot) has N non-trivial elements. On the

other hand, the complex (1.7) obtained from (4.2) has only two non-trivial elements,

0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 ⊖Q 0 0 0 0 0 0 0 Q 0 0

(4.3)

where we used the equivalence Q⊖Q ∼= 0.

17

4.2. The Trefoil Knot

In this case, the complex (1.7) has six non-trivial elements9. Similarly, there are only

six non-zero BPS invariants N ,Q,s for the following values of (Q, s):

(1,−2), (3,−2), (1, 2), (3, 0), (3, 2), (5, 0) (4.5)

It is natural to expect that each of the non-zero BPS invariants comes from a single

cohomology group which, in turn, corresponds to a particular element in (1.7) with the

same value of (Q, s):

N ,Q,s = (−1)rDQ,s,r (4.6)

Notice, there is no summation over r in this formula. There is a unique choice of such

DQ,s,r consistent with the sl(2) homological invariant (2.20):

D3,−2,0 = 1, D3,0,2 = 1, D5,0,3 = 1,

D1,−2,0 = −1, D1,2,2 = −1, D3,2,3 = −1(4.7)

Given these numbers, one can use (1.8) to compute all sl(N) cohomology groups for the

trefoil knot. For example, substituting (4.7) into (1.8) we find the following homological

sl(3) invariant:

KhN=3(31) = q2 + q4 + q6 + q6t2 + q8t2 + q12t3 + q14t3 (4.8)

Remarkably, this result is in perfect agreement with the one obtained using the technique

of foams10 [16]. For the sl(4) and sl(5) homology, we predict

KhN=4(31) = q3 + q5 + q7 + q9 + q7t2 + q9t2 + q11t2 + q15t3 + q17t3 + q19t3 (4.9)

9 For example, for N = 2 it has the form

3 | ⊖Q Q

2 | ⊖Q Q

1 |0 | ⊖Q Q

+ −0− −1− −2− −3− −4− −5− −6− −7− −8− −9− −10− → j

(4.4)

where the vertical (resp. horizontal) direction represents index i (resp. index j) and we used

Q⊖Q ∼= 0.10 We thank M. Khovanov for explanations and very helpful discussions on the “computational

shortcuts” used in these computations.

18

KhN=5(31) = q4+q6+q8+q10+q12+q8t2+q10t2+q12t2+q14t2+q18t3+q20t3+q22t3+q24t3

(4.10)

We note that, the structure of the refined integer invariants (4.7) for Q = 1 and Q = 5

has the form (3.8) consistent with a contribution of genus-g curves. For example, for Q = 1

we have ∑

s,r

DQ,s,rqstr = t

[− (q−1t−1/2 + qt1/2)2 + 2

](4.11)

which corresponds to a contribution of curves with g = 0 and g = 1. This is a strong

evidence suggesting that embedding the knot homology in our physical setup is indeed

correct. Moreover, since both terms in (4.11) have an extra factor of t, it also suggests,

though not conclusively, that these curves are isolated and that we indeed have an affine

change of basis in comparing knot homology with BPS spectrum, as discussed in the

previous section. It would be extremely interesting to consider the Lagrangian D-brane

associated to the trefoil and check whether or not there are isolated curves with Q = 1

and g = 0, 1 ending on it.

We find similar contributions of genus-g curves for other torus knots, T2,2m+1, where

the maximal genus of such curves grows linearly with the number of crossings, gmax = m.

Again, these examples suggest that, in order to compare the invariants DQ,s,r with the

physical BPS degeneracies DQ,s,r, one has to make a change of basis, r → r −Q/2. Note

that in the case of the unknot we did not have to change the basis. This suggests that the

change of basis involves a simple knot invariant αK , cf. (3.9).

4.3. The Knot 51

Following the same reasoning as in the previous example, we consider the set of values

(Q, s) which correspond to non-zero integer invariants N ,Q,s:

(3, 0), (3,−4), (3, 4), (5, 0), (5,−2), (5, 2), (5,−4), (5, 4), (7,−2), (7, 2)

(4.12)

Again, following the natural assumption that these are the values of (Q, s) which corre-

spond to non-trivial elements in (1.7), we can find the invariants DQ,s,r which encode the

sl(N) homological invariants,

D5,−4,0 = 1, D5,−2,2 = 1, D7,−2,3 = 1, D5,2,4 = 1, D7,2,5 = 1,

D3,−4,0 = −1, D3,0,2 = −1, D5,0,3 = −1, D3,4,4 = −1, D5,4,5 = −1(4.13)

19

Given these numbers, one can use (1.8) to compute all sl(N) cohomology groups for the

knot 51. For example, in the case N = 3 we get

KhN=3(51) = q6+q8+q10+q10t2+q12t2+q16t3+q18t3+q14t4+q16t4+q20t5+q22t5 (4.14)

Again, this result is in a perfect agreement with the sl(3) knot homology computed using

the technique of foams [16]. It would be interesting to check the other sl(N) knot homology

groups that follow from (4.13).

It is easy to generalize these examples to arbitrary torus knots T2,2m+1. For every

torus knot T2,2m+1, we find a complete agreement between our predictions based on (1.8)

and the sl(3) knot homology groups computed using the technique of web cobordisms (the

details will be discussed elsewhere [38]).

5. Generalizations

The proposed relation between the knot cohomology and the spectrum of BPS states

leads to non-trivial predictions for both knot theory and physics. Apart from the com-

putational predictions, examples of which we discussed in the previous section, there are

interesting generalizations suggested by this relation. For example, it suggests that there

should exist a categorification of more general knot invariants associated with arbitrary

representations of sl(N), not just the fundamental representation. Similarly, since knot co-

homology can be defined over arbitrary ground fields, including finite number fields, there

should exist corresponding physical realizations in terms of BPS states. This would be in-

teresting to study further. Also, it would be very exciting to find a combinatorial definition

of the integer invariants DQ,s,r, and of their generalizations to other representations.

The formulation of the sl(N) homological invariant in terms of the refined BPS in-

variants DQ,s,r can also be extended to links. Let L be an oriented link in S3 with ℓ

components, K1, . . . , Kℓ, all of which carry a fundamental representation of sl(N). As in

(1.1), the expectation value of the corresponding Wilson loop operatorW (L) = W ,···, (L)

is related to the polynomial sl(N) invariant, with a minor modification,

PN (L) = q−2N lk(L)〈W (L)〉 (5.1)

where lk(L) =∑

a<b lk(Ka, Kb) is the total linking number of L. The formulation of

PN (L) in terms of integer BPS invariants also needs some modification [8]. Namely, in our

notations,

〈W (L)〉(c) = (q−1 − q)ℓ−2∑

Q,s

N( ,···, ),Q,sqNQ+s (5.2)

20

where 〈W (L)〉(c) is the connected correlation function. For example, for a two-component

link, we have

〈W (L)〉(c) = 〈W (L)〉 − 〈W (K1)〉〈W (K2)〉 (5.3)

and

PN (L) = q−2N lk(L)[PN (K1)PN (K2) +

∑

Q,s

N( , ),Q,sqNQ+s

](5.4)

where PN (K1) and PN (K2) denote the sl(N) polynomials of the link components.

We wish to write a similar expression for the graded Poincare polynomial KhN (L)

in terms of integer invariants DQ,s,r(L). Because of the corrections involving the sl(N)

invariants of the sublinks, this formulation is less obvious than in the case of knots. For

example, for a two-component link L, we find, cf. (1.8),

KhN (L) = q−2N lk(L)[tαKhN (K1)KhN (K2) +

1

q − q−1

∑

Q,s,r∈ZZ

DQ,s,rqNQ+str

](5.5)

where α is a simple invariant of L, and the integer invariants DQ,s,r and N( , ),Q,s are

related as follows:

N( , ),Q,s−1 −N( , ),Q,s+1 =∑

r∈ZZ

(−1)rDQ,s,r (5.6)

Notice, this relation is quite different from what we had in the case of knots, cf. (1.9).

The first term in (5.5) is similar to the first term in (5.4). In order to understand

the structure of the second term, note that the factor (q−1 − q)ℓ−2 in (5.2) is a product

of two terms, (q−1 − q)−1 and (q−1 − q)ℓ−1, which have different origin and should be

treated differently. The first term, (q−1− q)−1, comes from the Schwinger computation [7]

and remains intact once PN (L) is lifted to the homological sl(N) invariant. On the other

hand, the factor (q−1 − q)ℓ−1 is similar to the contribution of a genus-g curve, (q−1 − q)2g,

discussed in section 3. In the homological sl(N) invariant this factor is replaced by a

polynomial expression in q±1 and t±1. Applying the same logic to (5.2) gives the second

term in (5.5).

Acknowledgments

We would like to thank D. Bar-Natan, R. Dijkgraaf, M. Gross, K. Intriligator, A. Ka-

pustin, M. Khovanov, A. Klemm, M. Marino, H. Ooguri, J. Roberts and D. Thurston for

21

useful discussions. S.G. would also like to thank the Caltech Particle Theory Group for

kind hospitality. The work of A.S. is supported by NSF grant DMS-0204927. This work

was conducted during the period S.G. served as a Clay Mathematics Institute Long-Term

Prize Fellow. S.G. is also supported in part by RFBR grant 04-02-16880. The work of

C.V. is supported in part by NSF grants PHY-0244821 and DMS-0244464.

Note added: A preliminary version of the present work was presented by one of us

(S.G.) at Caltech and UCSD seminars in October 2004. Since then, there have been some

interesting developments which provide additional support for the conjecture (1.8), see

e.g. [39] for independent computations of the Khovanov-Rozansky homology in some of

the examples considered here (see also [38]).

22

Appendix A. HOMFLY Polynomial for Some Knots

Let us list the polynomial invariant f (q, λ) for some simple knots:

f (31) =λq−2 + q2λ− λ3 − λ3q−2 − q2λ3 + λ5

q−1 − q

f (41) =λ−3 − λ−1q−2 − q2λ−1 + λq−2 + λq2 − λ3

q−1 − q

f (51) =λ3 + λ3q−4 + q4λ3 − λ5 − λ5q−4 − λ5q−2 − q2λ5 − q4λ5 + λ7q−2 + q2λ7

q−1 − q

f (61) =λ−3 + λ−1 − q−2λ−1 − q2λ−1 − λ+ λ3q−2 + q2λ3 − λ5

q−1 − q

According to (2.6), the polynomial f (q, λ) can be viewed as a generating function of the

BPS invariants N ,g,Q (or N ,Q,s). Notice, that the BPS invariants for the knots 31, 41,

and 61 obey a simple relation,

N ,g,Q(31)− N ,g,Q(41) + N ,g,Q(61) = N ,g,Q(unknot) (A.1)

Appendix B. sl(2) Knot Homology for Some Knots

Here, following [18], we list Khovanov’s sl(2) invariants, Kh2(L), for some simple knots:

Kh2(31) = q + q3 + q5t2 + q9t3

Kh2(41) = q + q−1 + q−1t−1 + qt+ q−5t−2 + q5t2

Kh2(51) = q3 + q5 + q7t2 + q11t3 + q11t4 + q15t5

Kh2(52) = q + q3 + q3t+ q5t2 + q7t2 + q9t3 + q9t4 + q13t5

Kh2(61) = 2q−1 + q−1t−1 + q−5t−2 + q + qt+ q3t+ q5t2 + q5t3 + q9t4

Kh2(71) = q5 + q7 + q9t2 + q13t3 + q13t4 + q17t5 + q17t6 + q21t7

Notice, that the sl(2) homological invariants for the knots 31, 41 and 61 are closely related:

Kh2(31)−Kh2(41) +Kh2(61) =q + q−1

+ q3(1 + t) + q5(t2 + t3) + q9(t3 + t4)(B.1)

The right-hand side of this expression evaluated at t = −1 givesKh2(unknot), in agreement

with (A.1).

23

References

[1] A. Schwarz, “New topological invariants arising in the theory of quantized fields,”

Baku International Topological Conf., Abstracts (part II) (1987).

[2] E. Witten, “Quantum Field Theory And The Jones Polynomial,” Commun. Math.

Phys. 121, 351 (1989).

[3] P. Freyd, D. Yetter, J. Hoste, W. Lickorish, K. Millett, A. Oceanu, “A New Polynomial

Invariant of Knots and Links,” Bull. Amer. Math. Soc. 12 (1985) 239.

[4] E. Witten, “Chern-Simons gauge theory as a string theory,” Prog. Math. 133 (1995)

637, hep-th/9207094.

[5] R. Gopakumar and C. Vafa, “On the gauge theory/geometry correspondence,” Adv.

Theor. Math. Phys. 3 (1999) 1415, hep-th/9811131.

[6] R. Gopakumar and C. Vafa, “M-theory and topological strings. I,II,” hep-th/9809187;

hep-th/9812127.

[7] H. Ooguri, C. Vafa, “Knot Invariants and Topological Strings,” Nucl.Phys. B577

(2000) 419.

[8] J. M. F. Labastida, M. Marino and C. Vafa, “Knots, links and branes at large N,”

JHEP 0011, 007 (2000), hep-th/0010102.

[9] S. Hosono, M.-H. Saito, A. Takahashi, “Holomorphic Anomaly Equation and BPS

State Counting of Rational Elliptic Surface,” Adv.Theor.Math.Phys. 3 (1999) 177.

[10] S. Hosono, M.-H. Saito, A. Takahashi, “Relative Lefschetz Action and BPS State

Counting,” Internat. Math. Res. Notices, (2001), No. 15, 783.

[11] S. Katz, A. Klemm and C. Vafa, “M-theory, topological strings and spinning black

holes,” Adv. Theor. Math. Phys. 3 (1999) 1445, hep-th/9910181.

[12] D. Maulik, N. Nekrasov, A. Okounkov, R. Pandharipande, “Gromov-Witten theory

and Donaldson-Thomas theory, I,” math.AG/0312059.

[13] S. Katz, “Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of

Calabi-Yau threefolds,” math.ag/0408266.

[14] M. Khovanov, “A categorification of the Jones polynomial,” math.QA/9908171.

[15] M. Khovanov, “Categorifications of the colored Jones polynomial,” math.QA/0302060.

[16] M. Khovanov, “sl(3) link homology I,” math.QA/0304375.

[17] M. Khovanov, L. Rozansky, “Matrix factorizations and link homology,” math.QA/0401268.

[18] D. Bar-Natan, “On Khovanov’s categorification of the Jones polynomial,” math.QA/0201043.

[19] M. Khovanov, “An invariant of tangle cobordisms,” math.QA/0207264.

[20] M. Jacobsson, “An invariant of link cobordisms from Khovanov’s homology theory,”

math.GT/0206303.

[21] A. Shumakovitch, KhoHo — a program for computing and studying Khovanov ho-

mology, http://www.geometrie.ch/KhoHo

24

[22] T. J. Hollowood, A. Iqbal and C. Vafa, “Matrix models, geometric engineering and

elliptic genera,” hep-th/0310272.

[23] J. M. F. Labastida and M. Marino, “Polynomial invariants for torus knots and topo-

logical strings,” Commun. Math. Phys. 217 (2001) 423, hep-th/0004196.

[24] J. M. F. Labastida and M. Marino, “A new point of view in the theory of knot and

link invariants,” math.qa/0104180.

[25] H. Murakami, T. Ohtsuki, S. Yamada, “HOMFLY polynomial via an invariant of

colored plane graphs,” Enseign. Math. 44 (1998) 325.

[26] P. S. Aspinwall, “D-branes on Calabi-Yau manifolds,” hep-th/0403166.

[27] M. Kontsevich, unpublished.

[28] A. Kapustin and Y. Li, “D-branes in Landau-Ginzburg models and algebraic geome-

try,” JHEP 0312 (2003) 005, hep-th/0210296.

[29] D. Orlov, “Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg

Orbifold,” math.AG/0302304.

[30] I. Brunner, M. Herbst, W. Lerche and B. Scheuner, “Landau-Ginzburg realization of

open string TFT,” hep-th/0305133.

[31] S. K. Ashok, E. Dell’Aquila and D. E. Diaconescu, “Fractional branes in Landau-

Ginzburg orbifolds,” hep-th/0401135.

[32] S. K. Ashok, E. Dell’Aquila, D. E. Diaconescu and B. Florea, “Obstructed D-branes

in Landau-Ginzburg orbifolds,” hep-th/0404167.

[33] K. Hori and J. Walcher, “F-term equations near Gepner points,” hep-th/0404196.

[34] I. Brunner, M. Herbst, W. Lerche and J. Walcher, “Matrix factorizations and mirror

symmetry: The cubic curve,” hep-th/0408243.

[35] “Mirror Symmetry” (Clay Mathematics Monographs, V. 1), K. Hori et.al. ed, Amer-

ican Mathematical Society, 2003.

[36] A. Schwarz and I. Shapiro, “Some remarks on Gopakumar-Vafa invariants,” hep-

th/0412119.

[37] C. Taubes, “Lagrangians for the Gopakumar-Vafa conjecture,” math.DG/0201219.

[38] N. Dunfield, S. Gukov, J. Rasmussen, “The Superpolynomial for Knot Homologies,”

math.GT/0505662.

[39] D. Bar-Natan, “Some Khovanov-Rozansky Computations” ;

http://www.math.toronto.edu/ drorbn/Misc/KhovanovRozansky/index.html

25